C. Algebra · Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling

Creating and Translating Patterns with Rational Numbers

C1.2 A1About 45 minutes

Learning goals

I can create repeating, growing and shrinking patterns that use fractions, decimals and integers.
I can show the same pattern in words, a table, a graph, an expression and an equation.
I can translate a linear pattern from one representation to another.

Words to know

rational number
Any number that can be written as a fraction of two integers (with a denominator that isn't 0). Integers, fractions, mixed numbers, and decimals that stop or repeat are all rational: 5, −3, 3/4, −2.5, 0.333...

representation
A way to show a pattern: objects, words, a table, a graph, an expression or an equation.

algebraic expression
A combination of numbers, variables and operations with no equal sign, like 2.5 − 1.5h.

equation
A statement that two things are equal, like T = 2.5 − 1.5h.

translate
To change a pattern from one representation to another, for example from a table to an equation.

Remember from Grade 7

In Grade 7 you created patterns with decimals and integers and showed them with tables, graphs, expressions and equations.

In Grade 8 you'll do the same with all rational numbers: fractions, mixed numbers, decimals and integers, positive or negative. A rational number is any number that can be written as one integer divided by another, like −5/2 or 3/4. Even 0.333... is rational, because it equals 1/3.

Let's start on a November evening

At 4 p.m. the temperature is 2.5°C. After sunset it drops 1.5°C each hour. Here is the same pattern shown five ways.

1. Words: Start at 2.5°C and subtract 1.5°C every hour.

2. Table:

h (hours after 4 p.m.)	0	1	2	3	4	5
T (°C)	2.5	1	−0.5	−2	−3.5	−5

The temperature drops below 0°C between 5 p.m. and 6 p.m.

3. Graph: the points fall 1.5 for each step to the right and lie on a straight line. The line crosses 0°C between h = 1 and h = 2.

4. Expression: 2.5 − 1.5h gives the temperature after h hours.

5. Equation: T = 2.5 − 1.5h connects the temperature T to the time h.

The expression is a recipe for a value. The equation says that T is equal to that value. Each representation is useful for something different:

the table for exact values,
the graph for seeing the trend and comparing,
the equation for any term, like h = 10: T = 2.5 − 15 = −12.5°C.

Translating tips

Table → equation: find the rate (difference) and the value at 0.
Equation → table: substitute n = 0, 1, 2, 3, ...
Graph → equation: read where the line crosses the vertical axis, and how much it rises or falls per step.
Words → equation: "start at" is the initial value; "add" or "subtract each time" is the rate.

Worked example

Example 1: Words to table to equation

Term 1 is 3/4, and you add 1/2 each time. Make a table and write an equation.

Terms: 3/4, 5/4, 7/4, 9/4 (adding 2/4 each time).
Rate: 1/2. Initial value (term 0): 3/4 − 2/4 = 1/4.
Equation: t = (1/2)n + 1/4.
Check term 3: 3/2 + 1/4 = 6/4 + 1/4 = 7/4 ✓.

t = (1/2)n + 1/4

Worked example

Example 2: Table to expression

A table shows n = 1, 2, 3, 4 and t = −2.4, −1.8, −1.2, −0.6. Write an expression for term n and find term 10.

Differences: +0.6 each time. Rate: 0.6.
Initial value: −2.4 − 0.6 = −3.
Expression: 0.6n − 3. Term 10: 0.6 × 10 − 3 = 6 − 3 = 3.

0.6n − 3; term 10 is 3.

Worked example

Example 3: Equation to graph

Graph y = −0.5x + 2 for x from −2 to 4.

Make a table: x = −2 → y = 1 + 2 = 3. x = −1 → 2.5. x = 0 → 2. x = 1 → 1.5. x = 2 → 1. x = 3 → 0.5. x = 4 → 0.
Plot each point and join them with a straight line, because x can be any number here.

The line starts at 2 on the y-axis and goes down 0.5 for each 1 to the right.

The graph of y = −0.5x + 2 for x from −2 to 4. Each point is 0.5 lower than the one to its left.

Worked example

Example 4: A repeating pattern with rational numbers

Make a repeating pattern whose core is 1/2, −1/4, 0. What is the 20th term?

The pattern is 1/2, −1/4, 0, 1/2, −1/4, 0, ...
The core has 3 terms. 20 = 6 × 3 + 2, so the 20th term is the 2nd term of the core.

The 20th term is −1/4.

Hands-on activity

Measuring cup patterns

You need: a measuring cup with markings (2-cup or 500 mL), a 1/4-cup and a 1/3-cup measure, water, a towel for spills.

Growing pattern:

Pour 1/2 cup of water into the big measuring cup. That's step 0.
Add 1/4 cup at each step and read the level. Record steps 0 to 4 in a table.
Write an equation: V = (1/4)n + 1/2. Predict the step when the water reaches 2 cups, then check.

Shrinking pattern:

Start with 2 cups. Scoop out 1/3 cup each step. Record the amounts: 2, 1 2/3, 1 1/3, 1, 2/3, ...
Write the equation V = 2 − (1/3)n and draw both patterns on one graph.

What to look for: the growing pattern reaches 2 cups at step 6 (1/2 + 6 × 1/4 = 2). The shrinking pattern is empty at step 6 (2 ÷ 1/3 = 6). The lines cross between step 2 and step 3 (where (1/4)n + 1/2 = 2 − (1/3)n, at n = 18/7, about 2.6).

Just for fun

A table, a graph and an equation walk into a café. The server takes one look and says, "Let me guess: you're all the same pattern, just dressed differently?"

Your turn

Write the first 4 terms: start at −5 and add 1.25 each time.

Describe t = 10 − 0.4n in words.

Make a table for t = (1/3)n, n = 1 to 4.

Write an equation for 0.2, 0.5, 0.8, 1.1, ... (term 1 is 0.2).

List the points to plot for y = 2x − 1 at x = −1, 0, 1, 2.

What is the difference between 4n − 2 and t = 4n − 2?

A repeating pattern has the core −3, 0, 3. What is the 17th term?

More practice: warm-up

Write the next term: 1/2, 1, 3/2, 2, ...

Write the next term: 0.6, 0.4, 0.2, 0, ...

Find t when n = 3 in t = 2n + 0.5.

Is −3/4 a rational number? Why?

What is the rate of −7, −5, −3, ...?

More practice: on your own

Make a table for t = −1.5n + 6 for n = 1 to 5. Which term is 0?

"Start at −8 (term 0) and increase by 2.5 each time." Write the equation and find term 6.

Write an equation for 5/6, 4/3, 11/6, 7/3, ... (term 1 is 5/6).

A graph shows the points (0, 4), (1, 3.25), (2, 2.5), (3, 1.75). Write its equation.

Create a linear shrinking pattern with rational numbers that reaches 0 at term 6. Give its equation.

Challenge

For t = 4 − (2/3)n, which is the first term that is negative?

A linear pattern has term 2 = 1.1 and term 5 = 2.3. Write its equation.

A 2.4 m ribbon has 0.15 m cut off for each bow. Write an equation for the length left. After how many bows is it first shorter than 1 m?

Check your understanding

Quiz

Which equation means "start at 6 (term 0) and subtract 0.5 each time"?

What is the next term of −2, −1.25, −0.5, ...?

Which is an expression, not an equation?

Which point is on the graph of y = −2x + 1?

What is the rate of 1/4, 1/2, 3/4, 1, ...?

A repeating pattern has the core 1/2, 1, −1. What is the 10th term?

For parents and teachers

Big idea: C1.2 asks Grade 8 students to create and translate patterns involving rational numbers (fractions, decimals and integers) using various representations, including algebraic expressions and equations for linear patterns.

Common difficulties:

Fraction arithmetic slowing everything down. Rewrite fractions with a common denominator (like sixths) before finding differences.
Plotting negative values. Count down from 0 on the vertical axis.
Mixing up expressions and equations. An equation has an equal sign.
Joining dots when the situation only has whole steps. Join the points with a line only if in-between values make sense (time, temperature), not for things like numbers of people.

How to help: Do the measuring-cup activity together and let your child choose the representation they find clearest, then ask them to translate it into one other form. This lesson builds A1 skills: representing and connecting ideas.